Theorem isumclim3 13897

Description: The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that j must not occur in A. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
isumclim3.1	|- Z = ( ZZ>= ` M )
isumclim3.2	|- ( ph -> M e. ZZ )
isumclim3.3	|- ( ph -> F e. dom ~~> )
isumclim3.4	|- ( ( ph /\ k e. Z ) -> A e. CC )
isumclim3.5	|- ( ( ph /\ j e. Z ) -> ( F ` j ) = sum_ k e. ( M ... j ) A )

Assertion

Ref	Expression
isumclim3	|- ( ph -> F ~~> sum_ k e. Z A )

Distinct variable groups:   A, j   j, k, M   ph, j, k   j, Z, k   j, F

Allowed substitution hints:   A( k)   F( k)

Proof of Theorem isumclim3

Dummy variables m x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumclim3.3	. . 3 |- ( ph -> F e. dom ~~> )
2		climdm 13695	. . 3 |- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
3	1, 2	sylib 201	. 2 |- ( ph -> F ~~> ( ~~> ` F ) )
4		sumfc 13852	. . . 4 |- sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = sum_ k e. Z A
5		isumclim3.1	. . . . 5 |- Z = ( ZZ>= ` M )
6		isumclim3.2	. . . . 5 |- ( ph -> M e. ZZ )
7		eqidd 2472	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
8		isumclim3.4	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> A e. CC )
9		eqid 2471	. . . . . . 7 |- ( k e. Z |-> A ) = ( k e. Z |-> A )
10	8, 9	fmptd 6061	. . . . . 6 |- ( ph -> ( k e. Z |-> A ) : Z --> CC )
11	10	ffvelrnda 6037	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
12	5, 6, 7, 11	isum 13862	. . . 4 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
13	4, 12	syl5eqr 2519	. . 3 |- ( ph -> sum_ k e. Z A = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
14		seqex 12253	. . . . . . 7 |- seq M ( + , ( k e. Z |-> A ) ) e. _V
15	14	a1i 11	. . . . . 6 |- ( ph -> seq M ( + , ( k e. Z |-> A ) ) e. _V )
16		isumclim3.5	. . . . . . 7 |- ( ( ph /\ j e. Z ) -> ( F ` j ) = sum_ k e. ( M ... j ) A )
17		fzssuz 11865	. . . . . . . . . . . . . 14 |- ( M ... j ) C_ ( ZZ>= ` M )
18	17, 5	sseqtr4i 3451	. . . . . . . . . . . . 13 |- ( M ... j ) C_ Z
19		resmpt 5160	. . . . . . . . . . . . 13 |- ( ( M ... j ) C_ Z -> ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A ) )
20	18, 19	ax-mp 5	. . . . . . . . . . . 12 |- ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A )
21	20	fveq1i 5880	. . . . . . . . . . 11 |- ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m )
22		fvres 5893	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
23	21, 22	syl5reqr 2520	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m ) )
24	23	sumeq2i 13842	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m )
25		sumfc 13852	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m ) = sum_ k e. ( M ... j ) A
26	24, 25	eqtri 2493	. . . . . . . 8 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ k e. ( M ... j ) A
27		eqidd 2472	. . . . . . . . 9 |- ( ( ( ph /\ j e. Z ) /\ m e. ( M ... j ) ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
28		simpr 468	. . . . . . . . . 10 |- ( ( ph /\ j e. Z ) -> j e. Z )
29	28, 5	syl6eleq 2559	. . . . . . . . 9 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
30		simpl 464	. . . . . . . . . 10 |- ( ( ph /\ j e. Z ) -> ph )
31		elfzuz 11822	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> m e. ( ZZ>= ` M ) )
32	31, 5	syl6eleqr 2560	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> m e. Z )
33	30, 32, 11	syl2an 485	. . . . . . . . 9 |- ( ( ( ph /\ j e. Z ) /\ m e. ( M ... j ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
34	27, 29, 33	fsumser 13873	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = ( seq M ( + , ( k e. Z |-> A ) ) ` j ) )
35	26, 34	syl5eqr 2519	. . . . . . 7 |- ( ( ph /\ j e. Z ) -> sum_ k e. ( M ... j ) A = ( seq M ( + , ( k e. Z |-> A ) ) ` j ) )
36	16, 35	eqtr2d 2506	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( k e. Z |-> A ) ) ` j ) = ( F ` j ) )
37	5, 15, 1, 6, 36	climeq 13708	. . . . 5 |- ( ph -> ( seq M ( + , ( k e. Z |-> A ) ) ~~> x <-> F ~~> x ) )
38	37	iotabidv 5574	. . . 4 |- ( ph -> ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x ) = ( iota x F ~~> x ) )
39		df-fv 5597	. . . 4 |- ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x )
40		df-fv 5597	. . . 4 |- ( ~~> ` F ) = ( iota x F ~~> x )
41	38, 39, 40	3eqtr4g 2530	. . 3 |- ( ph -> ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( ~~> ` F ) )
42	13, 41	eqtrd 2505	. 2 |- ( ph -> sum_ k e. Z A = ( ~~> ` F ) )
43	3, 42	breqtrrd 4422	1 |- ( ph -> F ~~> sum_ k e. Z A )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   C_ wss 3390   class class class wbr 4395   |-> cmpt 4454   dom cdm 4839   |` cres 4841   iotacio 5551   ` cfv 5589  (class class class)co 6308   CCcc 9555   + caddc 9560   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   seqcseq 12251   ~~> cli 13625   sum_csu 13829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830

This theorem is referenced by:  esumcvg  28981